Baesan Model Selecon for Srucural Brea Models * Andrew T. Levn Federal eserve Board Jere M. Pger Unvers of Oregon Frs Verson: Noveber 005 Ths verson: Aprl 007 Absrac: We ae a Baesan approach o odel selecon n regresson odels wh srucural breas n condonal ean and resdual varance paraeers. A novel feaure of our approach s ha does no assue nowledge of he paraeer subse ha undergoes srucural breas, bu nsead conducs odel selecon jonl over he nuber of srucural breas and he subse of he paraeer vecor ha changes a each brea dae. Sulaon experens deonsrae ha conducng hs jon odel selecon can be que poran n pracce for he deecon of srucural breas. We appl he proposed odel selecon procedure o characerze srucural breas n he paraeers of an auoregressve odel for pos-war U.S. nflaon. We fnd poran changes n boh resdual varance and condonal ean paraeers, he laer of whch s revealed onl upon conducng he jon odel selecon procedure developed here. Kewords: Poseror Model Probabl, Marov Chan Mone Carlo, Inflaon Perssence JEL Codes: C, C, C, C5 * Levn: Federal eserve Board, Sop 70, Washngon, DC 055 (Andrew.levn@frb.org); Pger: Dep. of Econocs, 85 Unvers of Oregon, Eugene, O 97403-85 (jpger@uoregon.edu). We han Me McCracen, Jaes Morle, Jonahan Wrgh, and senar parcpans a he Unvers of Oregon for helpful coens. The vews expressed n hs paper are solel he responsbl of he auhors, and should no be nerpreed as reflecng he vews of he Board of Governors of he Federal eserve Sse.
. Inroducon egresson odels wh paraeers ha undergo srucural change have becoe a saple of he appled e-seres econoercan s ool. One reason for he popular of such odels s he subsanal evdence for paraeer nsabl n regressons nvolvng e econoc varables, parcularl acroeconoc and fnancal varables, easured over he pos-war saple perod. For exaple, here s overwhelng evdence of paraeer breas n auoregressve odels of U.S. real oupu, parcularl n he resdual varance paraeer (K and Nelson, 999 and McConnell and Perez-Quros, 000). Lewse, Garca and Perron (996), apach and Wohar (005) and Levn and Pger (00) fnd poran shfs n he nercep paraeer of auoregressve odels for neres raes and nflaon n G7 counres. Indeed, Soc and Wason (996) docuen nsabl n he paraeers of a unvarae auoregresson for a sgnfcan fracon of he 76 U.S. acroeconoc e-seres he sud over he pos-war perod. In an cases, he naure of srucural breas, n ers of her nuber and ng, s no nown ex-ane, and a large leraure has eerged focusng on esng for he exsence of srucural breas, where he dae of he poenal srucural brea s unnown. For exaple, Andrews (993), Andrews and Ploberger (994) and Debold and Chen (996) develop ess of a odel wh no srucural breas agans he alernave of a odel wh a sngle srucural brea. Ba and Perron (998) and Ba (999) develop sequenal esng procedures desgned o reveal he nuber of, perhaps ulple, srucural breas. Wang and Zvo (000) dscuss Baesan esaon of a e-seres odel wh ulple srucural breas, as well as presen approaches, based on Baesan odel coparson, o deerne he nuber of srucural breas.
In developng hese approaches o deerne he nuber of srucural breas, he leraure has aen he subse of paraeers ha change a each brea dae as gven. Tha s, here has been lle aenon pad o he odel selecon queson of whch paraeers change a each brea dae. Ths s an poran osson, as here are reasons o beleve ha a be poran o jonl conduc odel selecon over he nuber of srucural breas and he paraeers ha change a each brea dae. Perhaps os poranl, evdence for a brea a be revealed onl f he subse of paraeers ha undergo srucural breas s correcl specfed. For exaple, n evaluang he evdence for paraeer breas, suppose he researcher has no a pror nowledge of whch paraeers are lel o have undergone breas, and hus allows all paraeers o change a each brea dae, a coon pracce n esng for srucural breas. Such a procedure s lel o have low power o denf srucural breas f onl a sall subse of he paraeer vecor acuall changes. Furher, even f one s able o accurael deerne he nuber of srucural breas, nerpreng he econoc eanng of he breas a be aded b denfng whch paraeers brea a each brea pon. For exaple, n e seres odels of acroeconoc varables, he econoc nerpreaon of changes n he perssence of he seres s ofen que dfferen fro he econoc nerpreaon of changes n he resdual varance. I s no dffcul o fnd exaples of regresson odels where careful aenon o esablshng he subse of he paraeer vecor ha undergoes paraeer change gh eld poran dvdends. For exaple, a lvel debae has eerged on he exsence of shfs n condonal ean paraeers of e equaons for odels of he U.S. acroecono, such as he Phllps Curve and he Federal eserve s reacon funcon.
On he one hand, Clarda, Gal and Gerler (000), Cogle and Sargen (00, 005), and Bovn (999) fnd poran whn-saple varaon n her esaes of e condonal ean paraeers. However, Ss (999, 00) and Ss and Zha (004) argue ha allowng for such changes does no provde a sascall superor f over odels wh consan condonal ean paraeers and srucural change n covarance arx paraeers. Gven he poenall large nuber of condonal ean paraeers ha a change n such odels, he resuls could be que sensve o wheher all condonal ean paraeers are allowed o change, or onl a subse. Indeed, Bovn (999) noes ha a e reason for dscrepances n sascal evdence for srucural change observed n hs leraure s he dfferng nuber of paraeers ha are allowed o brea n alernave odel specfcaons. In hs paper we ae a Baesan approach o odel selecon n regresson odels wh srucural breas n condonal ean and resdual varance paraeers. An poran eleen of our approach s ha does no condon on he paraeer subse ha undergoes srucural breas, bu nsead conducs odel selecon jonl over he nuber of srucural breas and he subse of he paraeer vecor ha changes a each brea dae. Specfcall, we proceed b copung and coparng poseror odel probables, where he space of poenal odels s expanded o nclude odels ha dffer no jus b he nuber of srucural breas, bu also b whch eleens of he paraeer vecor are fxed and whch eleens change across ndvdual brea daes. The Baesan approach s well sued for he odel selecon proble suded n hs paper, as hs proble nvolves he coparson of non-nesed odels. For exaple, one a be neresed n coparng a odel n whch here are wo srucural breas, one n 3
he nercep paraeer and one n he resdual varance paraeer, agans a odel n whch here are wo breas n he resdual varance paraeer. The Baesan approach proceeds b coparng poseror probables for varous copeng odels, an approach for whch non-nesed odels creae no specal consderaons. Baesan odel selecon for he nuber of srucural breas was developed and dscussed b Wang and Zvo (000). Here we exend he Wang and Zvo fraewor o allow for odel selecon ha encopasses he subse of paraeers ha undergoes srucural breas. To evaluae he perforance of our proposed odel selecon procedure, we conduc a seres of sulaon experens n whch we generae arfcal daa fro regresson odels wh varng nubers of srucural breas, and he srucural breas occur n a subse of he paraeer vecor. The resuls of hese experens sugges here are poenall szeable gans o conducng odel selecon over he subse of paraeers ha undergo breas raher han spl allowng all paraeers o change. In parcular, he lelhood of selecng he odel wh he correc nuber of srucural breas s subsanall enhanced when odel selecon s expanded o nclude he subse of paraeers ha undergo breas. Furher, he sulaon experens sugges ha he proposed Baesan approach s relavel successful a denfng he correc subse of paraeers ha undergo change. Fnall, we appl our odel selecon approach o characerze possble srucural breas n condonal ean and resdual varance paraeers n an auoregresson for he U.S. nflaon rae. There s subsanal ongong debae abou he exsence of such breas, wh Cogle and Sargen (00, 005) argung ha he nflaon process has Suers (004) uses he Wang and Zvo (000) ehodolog o odel srucural breas n OECD uneploen raes. 4
undergone poran changes n hese condonal ean paraeers, ncludng he perssence of he process, whle oher auhors, such as Soc (00) and Pvea and es (00) argue ha, whle here s srong evdence of changes n resdual varance paraeers, changes n condonal ean paraeers are uch less obvous. Our resuls reveal several nsghs ha conrbue o hs leraure. Frs of all, he Baesan odelselecon procedures sugges here have been subsanal changes n boh condonal ean and resdual varance paraeers. Second, and poranl, he evdence for condonal ean paraeers s revealed onl when one conducs odel selecon over he subse of condonal ean paraeers ha undergo breas, a resul ha deonsraes he eprcal relevance of our proposed odel selecon procedure. Fnall, he resuls sugges ha evdence for nercep paraeer shfs s subsanal, bu evdence for shfs n he perssence of he process s less so. Indeed, esaes of nflaon perssence obaned b Baesan odel averagng of he varous odels under consderaon suggess ha nflaon perssence has been roughl consan over he saple, albe a a subsanall lower level han esaes obaned assung a consan paraeer auoregresson. Secon las ou he eprcal odel of neres n hs paper and descrbes he Baesan approach o conducng odel selecon jonl over boh he nuber of srucural breas and he subse of paraeers ha undergo breas a each brea dae. Secon 3 deals he resuls of sulaon experens desgned o evaluae he perforance of he Baesan echnques. Secon 4 presens an applcaon of he Baesan odel selecon procedures o odelng U.S. nflaon raes. Secon 5 concludes and offers soe drecons for fuure research. 5
. Model Specfcaon, Baesan Esaon, and Model Selecon. Model Specfcaon Consder he followng e-seres regresson wh srucural breas:.. x β ε ' x β ε ' x β ' ε τ τ, τ τ,.. 0 τ τ, (),...,T, where ( ) s a scalar dependen varable observed a e, ( x x x ) ' x,..., s a,,,, β β β s a ( ), β,..., vecor of exogenous or predeerned covaraes, ( ) ' vecor of coeffcens,,...,, and ε ~.. d. N(0,). The nal and fnal brea daes, τ 0 and τ are equal o 0 and T respecvel. The reanng brea daes are assued unnown and reaed as addonal paraeers o be esaed. The odel n () can be cas n arx noaon as follows. Defne ( T ( ) ) arx whose ( ) oherwse. In oher words, D as he, eleen s equal o one f τ < τ and zero D s a arx whose h colun s a du varable ndcang hose e perods ha are ncluded n he rege begnnng a τ and endng a τ. Collec he condonal ean and resdual varance paraeers no he vecors β ( β, β,..., β, β, β,..., β β ) ' and (,,..., ) ',...,. 6
~, ~ x ( x ) ',, x,,..., x,, j,...,, and ~ s. We,,..., T Fnall, defne ( ) ' can hen wre equaon () as: j j j j T D ~ Xβ u~, () where X [ D ~ x D ~ x.. D ~ x ] s a ( T (( ) * ) ) arx wh ndcang eleen b eleen ulplcaon, and u ~ ~ N(0, I ~ s T T ). We wan o consder odels where no all paraeers are allowed o change across each brea dae. Defne he arx hrough pos-ulplcaon of β, j,..., j ( q ), as he ( ) D, sus he h and h coluns of arx ha, β j D f β, j β j wh defned analogousl. For exaple, for a odel wh consan resdual varance paraeer, he arx s spl an (( ) ) arx of ones. Alernavel, for a odel wh resdual varance paraeer ha s allowed o change across all brea daes, s he ( ) den arx. In hs noaon, q β and q j denoe he nuber of reges over whch he coeffcen on x~ j and he resdual varance paraeer are allowed o ae on dfferen values respecvel. Thus, he oal nuber of unque condonal ean paraeers s qβ and he oal nuber of unque resdual varance j paraeers s q. Noe ha j β and j arces ha resrc all paraeers o be equal ' across an parcular brea dae, ha s ha resrc ( ) ( ) ' β, nadssble, as hese resrcons pl ha he nuber of srucural breas s no, bu -. β,, are 7
Defne he vecor β as he arx β reduced o conan onl he j qβ unque j eleens of β, and as he vecor reduced o conan onl he q unque eleens of. The odel n () can hen be rewren o ncorporae cross-rege paraeer equal resrcons as follows: ~ X β u~, (3) where X [( D ) ~ x ( D ) ~ x.. ( D ) ~ x ] β β β, and u ~ N(0T, IT s ) ~ s D. wh ( ) ~ ~. Pror Specfcaon In hs paper we focus on Baesan esaon of he odel n (3). We begn wh specfcaon of pror dens funcons for he odel paraeers, condonal on values for and {,...,, }. Paron he paraeers no hree blocs, gven b β, β β, and ( ) ' τ τ. We assue pror ndependence of β,, τ,..., τ, and τ, as well as pror ndependence of he eleens of. The jon pror s hen gven b: f ( β τ, ) f ( β) f ( τ ) q, f ( ). (4) We specf proper prors for each paraeer bloc. We defne f ( β ) as a ulvarae Gaussan rando varable wh ean vecor C and varance-covarance arx Σ. For each,,...,q, we specf ( ) f as an nvered gaa dens 8
funcon wh paraeers ν and δ. Fnall, our pror for τ s a unfor dsrbuon over he space of allowable brea daes. A se of brea daes s allowable f he ee he followng se of resrcons: ( τ ) b, j τ, (5) j > where b. The consran (5) requres ha a rege have nu lengh b..3 Baesan Esaon va he Gbbs Sapler For gven values of and, and gven he pror dens funcons defned above, Baesan esaon of he odel n (3) can proceed va he Gbbs Sapler (Gelfand and Sh, 990). The Gbbs sapler for a e-seres regresson odel wh ulple srucural breas s descrbed n Wang and Zvo (000). The algorh presened here follows closel ha of Wang and Zvo (000), odfed o allow for subses of paraeers ha brea across soe, bu no all, brea daes. The Gbbs saplng algorh proceeds b drawng eravel fro he full se of condonal poseror denses n he followng seps: A) Generae a draw of β, denoed B) Generae a draw of, denoed β, fro f ( β, τ, ) ~, fro f ( β, τ, ~ ) C) Generae a draw of τ, denoed τ, usng he followng seps: C) Generae a draw of τ, denoed C) Generae a draw of τ, denoed τ, fro f ( τ ~ β,, τ,..., τ, ) τ, fro f ( τ ~ β,, τ, τ 3,..., τ, ) See also Sephens (994). 9
.. C) Generae a draw of τ, denoed D) Se and τ τ and repea seps A-D. τ, fro f ( τ β, ~, τ,..., τ, ) The algorh s naed wh arbrar values of and τ. Assung ceran regular condons are e (Terne, 994) draws fro hs algorh wll converge o draws fro he poseror dens of neres f ( β,, τ ) pon esaes and hghes poseror dens nervals. f ~, whch can be used o for Gven he Gaussan lelhood funcon and conjugae prors, draws fro ( β, τ, ) and f ( β, τ, ) ~ ~ are sraghforward. Specfcall, condonal on τ, he odel n (3) s a lnear regresson odel wh du varables. The condonal poseror dens funcon for β s hen: N f β ~ (6) (, τ, ) ~ ' ( ( ~ ' ' ) ) ( ( ~ ) ~ ), ( ( ~ Σ X IT s X Σ C X IT s Σ X IT s ) X ) ), fro whch saples can be easl obaned va draws fro a ulvarae rando noral q dens. I can also be shown ha ( ) f β, τ, ~ f ( β, τ, ~ ) f, and: ( β τ ~ ) IGν τ τ δ ( ~ ',, ~, X β )( ~ X β ) ( ), (7) where ~ and fro f ( β, τ, ) X hold he τ hrough τ rows of ~ and ~ s hen generaed as q draws fro f ( β, τ, ~ ) X respecvel. A draw,,...,q. 0
f To coplee he Gbbs-saplng algorh, we requre draws fro ( τ β,, τ, ) f ~, whch s gven b: ( τ β,, τ, ) L ~ ~. (8) ( τ β,, τ, ) f ( τ β,, τ ) f ( ~ β,, τ ) Gven he unfor pror, f ( τ ) on τ we have:, and he fac ha he denonaor of (8) does no depend f ( τ β,, τ ~ ) L( ~, τ β,, τ, ). (9) Expresson (9) can be splfed furher b nong ha: ( β,, τ, ~ ) L( τ β,, τ, τ, ) L τ. (0) τ,..., τ Thus, he condonal poseror dsrbuon funcon for τ s a ulnoal dsrbuon defned over he adssble range for τ, gven b τ b,...,τ b, wh probables proporonal o he lelhood funcon for τ evaluaed usng onl daa fro τ τ.,..., Noe ha seps C-C n he Gbbs Sapler above could be replaced wh a sngle draw fro f ( τ β,, ) ~, for whch s sraghforward o show: f τ ~ ~. () ( β,, ) L( τ β,, ) However, n pracce, obanng a draw fro () can be ver copuaonall nensve, as requres evaluang he lelhood funcon a all adssble τ. For exaple, for
T00, 3 and b, each draw fro () requres over llon evaluaons of he lelhood funcon. B conras obanng a draw of τ fro (9) requres less han evaluaons of he lelhood funcon. 3 * T.4 Model Selecon In pracce, he nuber of breas,, as well as he subses of paraeers ha change across each brea dae, defned b, are unnown. In he followng, we defne a odel as gven b he values of and, and denoe hs odel as M (, ). The proble of choosng and s hen cas n ers of odel coparsons across alernave ( ) M,. The sandard Baesan approach o odel coparson s o copue poseror probables of alernave odels. In parcular, he poseror probabl of M (, ) s gven b: ( ~ M (, ) ) P( M (, ) ) ( ~ f P M (, ) ) f ( ~. () ) In (), P ( M (, ) ) s he researcher s pror probabl on ( ) ( M ( ) ) M,, whle f ~, s he argnal lelhood, or lelhood funcon negraed free of odel paraeers. Fnall, f ( ~ ) s an negrang consan ha can be recovered, gven P ( M (, ) ) and f ( ~ M (, ) ) fro he consran ha ( M ( ) ~ ) P,. Obanng P ( M (, ) ~ ) requres nowledge of f ( ~ M (, ) ) and ( M ( ) ) P,. To oban he argnal lelhood, we use he approach of Chb (995), whch obans a 3 Chb (998) develops an alernave, copuaonall effcen, approach o draw he enre vecor of brea daes sulaneousl, whch s based on odelng he srucural brea process as an rege Marov-swchng process wh absorbng saes.
sulaon conssen esae of f ( M (, ) ) ~ based on he oupu of full and reduced runs of he Gbbs sapler descrbed n Secon.3. We have consdered alernave approaches o esang f ( M (, ) ) ~, such as hose based on porance saplng, and have found ha he approach of Chb (995) perfored bes n he sulaon evdence presened n Secon 3. To specf he pror odel probabl, we use a fla pror over he nuber of srucural breas up o a pre-specfed axu, denoed *, as well as a fla pror over he dfferen peruaons of consdered for a gven value of, denoed b s: N. Tha P ( *. (3) * ( M, ) ) N The frs er n (3) dvdes he probabl space equall aong he * poenal brea odels, * 0,,..., whle he second er dvdes he probabl space for a gven equall aong he Gven ( M ( ) ) N poenal odels. P, ~, selecon of and proceeds n wo seps. In he frs sep, we choose he nuber of srucural breas as he value of ha solves: ( ax ( P ( M ( ) ~ )) ax P( M (, ) ~ ). (4) * * 0,,.., ) ( 0,,.., ) Gven a choce for, n he second sep we choose as ha value of ha solves: ax P ( ) ( M (, ) ) ~. (5) 3
An alernave, coonl used approach o odel selecon s based on nforaon crera such as he Schwarz Inforaon Creron (SIC). The SIC was shown o be a conssen creron for selecng he nuber of srucural breas n a lnear regresson wh exogenous regressors b Lu e al. (997), and was shown o perfor well a selecng he nuber of srucural breas n dnac odels b Wang and Zvo (000). Here we wll also consder odel selecon based on he SIC. In parcular, he SIC for M (, ), denoed SIC (, ), s gven b: ( ˆ ~, ˆ, ˆ τ ) q q ( T ) SIC(, ) ln L β β ln j, (6) j where has ndcae he axu lelhood esaes. Due o he subsanal copuaonal burden nvolved wh obanng axu lelhood esaes for ceran srucural bea odels, we follow Wang and Zvo (000) and nsead evaluae (6) a he edan of he poseror dsrbuon for each paraeer. A value for and s hen chosen as he soluon o he followng proble: n * ( 0,,.., ; ) SIC ( M (, ) ~ ). (7).5 Model Averagng In an cases, he objecve s no o selec a parcular ( ) M, fro he se of possble srucural brea odels, bu nsead o draw nference on a parcular subse of he paraeer space ha has a coon nerpreaon across odels wh dfferen and. A dsnc advanage of he Baesan approach s ha allows he researcher o oban 4
a poseror dsrbuon for hs subse of paraeers of neres, whou condonng on a parcular odel. * Specfcall, suppose one s neresed n a subse of he paraeer space, denoed θ. * The poseror dens for θ, condonal on onl he observed daa, s gven as: (, ) * where M (, ) * ( M (, ), ~ )* P( M (, ) ~ ) * p( θ ~ ) f θ, (8) f θ ~ s he poseror dens of * θ condonal on a parcular odel, and can be sapled usng he algorh gven n secon.3, whle ( M ( ) ) P, ~ s he poseror odel probabl copued as n secon.4. Operaonall, a draw can be obaned fro ( * ~ ) (, ) * p θ b obanng a draw fro M (, ) f θ ~ for all and, and hen forng a weghed average of hese draws, where he weghs are gven b ( M ( ) ) P, ~..6 Densonal of he Model Space A praccal proble wh he odel selecon procedure oulned above s he prolferang densonal of he odel space. For exaple, consder a sple A() odel wh hree paraeers, nael an nercep, an auoregressve paraeer, and he resdual varance paraeer. These paraeers elds seven poenal cobnaons of paraeers ha can go undergo a srucural brea a an parcular brea pon. If he odel s allowed o have breas, and one wans o copare all possble cobnaons of paraeer change for hs brea odel, we have N 7 poenal odels o 5
consder. The odel space ncreases even furher f we copare across alernave values of. Tha beng sad, gven odern copung speeds, s que feasble o conduc odel selecon over all poenal odels for oderael paraeerzed regressons, such as unvarae auoregressons, and for oderae nubers of breas. Second, for ore hghl paraeerzed odels, such as a vecor auoregresson, odel selecon gh be conduced over paraeer blocs, whch would reduce he densonal of he odel space consderabl. Fnall, alhough consderng a large nuber of poenal odels has a easurable cos, he benef gh also be subsanal. Indeed, we wll deonsrae n he followng secons ha basng nference regardng srucural breas on a odel wh breas n all paraeers can lead o ver sleadng resuls, even for odels wh a sall nubers of paraeers. 3. Sulaon Evdence In hs secon we descrbe he resuls of sulaon experens conduced o evaluae he perforance of he odel selecon procedures dealed above. In hese sulaon experens we generae arfcal e seres fro regresson odels wh varng nubers of srucural breas n he condonal ean and condonal varance paraeers, and evaluae he abl of he Baesan and SIC-based odel selecon procedures o selec he correc nuber of breas and he correc subse of paraeers ha brea a each brea dae. The deals of hs sulaon experen are as follows. Gven a e seres of daa, ~, we focus on selecng a odel fro he followng class of odels: 6
.. β β x 0 0 0 β β x β β ε x ε ε τ τ, τ τ,.. 0 τ τ, (9),...,T, where ε ~.. d N(0,), x s scalar and follows a frs-order auoregressve process wh sandard noral nnovaons and auoregressve paraeer ρ, and he saple sze s se o T 00. We consder a axu nuber of srucural breas, *, equal o wo. We hen conduc odel selecon over all possble nubers of srucural breas,, and varaons of β, 0 β and. Ths elds a oal of 57 odels o copare, one correspondng o he 0 case, seven correspondng o he case, and 49 correspondng o he case. For each ( ) M,, we copue he poseror probabl of he odel, P ( M (, ) ), as well as he SIC, ( M ( ) ) odel as descrbed n Secon.4. SIC,, and selec a As a eans of coparson, we also consder he abl of a procedure ha does no conduc odel selecon over o selec he correc nuber of srucural breas. In hs procedure, onl hree poenal odels are copared, one for each value of under consderaon. The odel for each ses β, 0 β and equal o he denf arx, hus allowng each of β 0, β and o brea a each of he brea daes. A value for s hen chosen as he correspondng o he odel wh he hghes poseror probabl, where we agan specf a unfor pror over. We also consder he abl of he nu SIC obaned across odels wh alernave o selec he correc value 7
of. We refer o hese procedures ha do no conduc odel selecon over as he baselne poseror odds and baselne SIC procedures n he followng, whle he procedures ha do conduc odel selecon over are referred o as he preferred poseror odds and preferred SIC procedures. For Baesan esaon of each odel, ( ) M,, we use he followng ses of prors. The pror ean and varance-covarance arx for β, gven b C and Σ, are se equal o a vecor of zeros and he denf arx respecvel, plng ha each eleen of β has a sandard noral pror dsrbuon and s ndependen of all oher eleens of β. The pror for each resdual varance paraeer,, s nvered gaa wh paraeers ν. 00 and δ. Fnall, he nu rege lengh consdered s se o b, whch s 6% of he saple sze of T 00. esuls are based on 0,000 sulaons of he Gbbs Sapler afer an nal 5000 burn-n sulaons o oban convergence. In he frs se of sulaons, we evaluae he perforance of he odel selecon procedures when he rue odel has no srucural breas. In parcular, we generae daa fro (9) wh 0. I s well nown ha he perforance of frequens-based ess for srucural breas s que sensve o he perssence of regressors, wh ess beng severel overszed for hgh levels of perssence (Debold and Chen, 996). To evaluae he sensv of he procedures developed here o perssen regressors we consder hree possble calbraons of ρ, correspondng o low, oderae and hgh perssence, and gven b ρ 0. 3, ρ 0. 6 and ρ 0. 9. For each value of ρ, we se β β. The resuls repored below are based on 500 sulaons. 0 8
Table records he proporon of sulaons for whch he ndcaed odel selecon procedure chose he 0 odel. There are several es of parcular neres n hese resuls. Frs, he preferred poseror odds procedure perfors que well when he rue daa generang process has consan paraeers, selecng an (ncorrec) odel ha ncludes a srucural brea n onl abou % of he sulaons. For he preferred SIC procedure hese proporons are lower, bu sll above 90%. Second, he endenc for each of he odel-selecon procedures o falsel denf a srucural brea s largel unaffeced b he perssence of he regressor n he sulaed daa. Fnall, he perforance of he preferred poseror odds procedure s slar o ha for he baselne poseror odds procedure. Tha he preferred procedure does no prove on he baselne procedure n hs case s no surprsng, as here s no reason o expec ha conducng odel selecon over boh and would have an advanage n he case where he rue odel does no conan a srucural brea. However, here does no appear o be an noceable dsadvanage o usng he preferred approach o odel selecon eher. In he second se of sulaons, we evaluae he perforance of he odel selecon procedures when he rue odel has a sngle srucural brea. In parcular, we generae daa fro he odel n (9) wh. We consder sx poenal daa generang processes. In he frs wo, we generae daa fro a odel wh a brea n β 0 onl. We consder boh sall and large breas, where hese brea szes are calbraed b selecng values of β 0 and β 0 for whch he baselne poseror odds procedure selecs he correc value of approxael 5% and 75% of he e respecvel. 4 b { β 0, β 0.4} for he sall brea case and { 0, β 0.63} 0 0 0 0 These values are gven β for he large 4 Specfcall, he percenage of sulaons for whch he baselne Baesan procedure selecs he correc value for s whn one percenage pon of 5% for he sall brea sze and 75% for he large brea sze. 9
brea case. For each case, we se β. In he second se of daa generang processes, we consder a odel wh a brea n β onl. The brea s agan calbraed as beng eher sall or large as descrbed above, and paraeerzed as { β 0, β 0.3} for he sall brea case and { β 0, β 0.53} for he large brea case. For each case, we se β. Fnall, we consder wo cases for whch here s a brea n onl. 0 for he We agan consder sall or large breas, paraeerzed as { 0.73, } sall brea case, and { 0.63, } for he large brea case. For each case, we se β 0 β. For all cases, we se he brea dae, τ, n he ddle of he saple perod, so ha τ 00. The auoregressve paraeer for he regressor, ρ, s se equal o 0.6. We agan base our resuls on 500 sulaed e seres fro each daa generang process. Table records he proporon of sulaons for whch he ndcaed odel selecon procedure chose he odel. For he preferred poseror odds and SIC procedures, he able also records he proporon of sulaons for whch he odel selecon procedure chose boh and correcl. A prar resul o ephasze fro Table s ha conducng odel selecon over boh and elds subsanal proveens, n ers of he frequenc wh whch he correc value of s chosen, over he baselne procedures conduced over onl. For he sall brea case, where he baselne poseror odds procedure selecs he correc value of n 5% of he sulaons, he preferred poseror odds procedure selecs he correc value of n well above 40% of he sulaons n all cases. Ths proveen s even ore subsanal for SIC-based odel selecon. In hs case, he baselne procedure selecs he correc value of n 0% or 0
less of he sulaons, whle he preferred procedure selecs he correc value of n around 50% of he sulaons. No surprsngl, he proveens generaed b he preferred procedure becoe saller when we consder larger breas. However, he proveen s sll subsanal, on he order of 0-5 percenage pons for he poseror odds procedures and 0-5 percenage pons for he SIC procedures. Table also deonsraes ha he preferred procedures selec he correc value of boh and close o as ofen as he selecs he correc value of, suggesng ha he procedure s relavel successful a denfng he correc subse of paraeers ha brea a each brea dae. In he fnal se of sulaons, we evaluae he perforance of he odel selecon procedures when he rue odel has wo srucural breas. In parcular, we generae daa fro he odel n (9) wh. We agan consder sx poenal daa generang processes. In he frs wo, we generae daa fro a odel wh wo breas n β 0 onl. We agan consder boh sall and large breas, calbraed as dscussed above for he case 3 of a sngle srucural brea. These values are gven b { 0, β 0.67, β 0} 3 sall brea case and { 0, β 0.9, β 0} 0 0 0 β for he 0 0 0 β for he large brea case. For each case, we se β. For he second se of daa generang processes, we consder a odel wh wo breas n β onl. Here, sall and large breas are paraeerzed as { β 0, β 0.53, β 0} and { 0, β 0.74, β 0} β respecvel. For each case, we se β. Fnall, we consder wo cases for whch here are wo breas n 0 3 onl, wh sall and large breas paraeerzed as { 0.6,, 0.6} 3 { 0.5,, 0.5} and respecvel. For each case, we se β β. For all 0
sulaons, we space he brea daes, τ and τ, equall hroughou he saple a daes 67 and 34 respecvel. The auoregressve paraeer for he regressor, ρ, s se equal o 0.6. We agan base our resuls on 500 sulaed e seres fro each daa generang process. Table 3 records he proporon of sulaons for whch he ndcaed odel selecon procedure chose he odel. For he preferred poseror odds and SIC procedures, he able also records he proporon of sulaons for whch he odel selecon procedure chose boh and correcl. The resuls n Table 3 gve a slar essage o hose n Table for he case of a sngle srucural brea. Specfcall, n all cases consdered, he preferred poseror odds procedure selecs he correc value of subsanall ore han he baselne procedure. Ths proveen s even larger han was he case for a sngle srucural brea, whch s no surprsng gven ha he nuber of unnecessar paraeer breas allowed b he baselne procedure grew n he sulaons wh wo breas fro hose wh a sngle brea. Slar resuls are also obaned for he SIC procedures, alhough, agan, he benefs o conducng odel selecon over n addon o are ore pronounced. Fnall, Table 3 agan deonsraes ha he preferred procedures are farl successful a selecng he correc value of boh and, and hus he correc subse of paraeers ha brea a each brea dae. In suar, he resuls fro hese sulaon exercses are suggesve ha odel selecon conduced over boh he nuber of srucural breas and he subse of paraeers ha change a each brea dae can eld poran benefs over odel selecon procedures n whch all paraeers are allowed o brea a each brea dae. One such benef s an proved frequenc wh whch he correc nuber of srucural
breas are chosen, whch, as would be expeced, s parcularl he case as he oal nuber of poenal paraeer breas grows relave o he oal nuber of acual paraeer breas. Anoher benef s ha he odel selecon procedures conduced over boh and provde soe relable nforaon regardng he subse of paraeers ha undergo breas a each brea dae, nforaon ha s absen fro procedures ha spl allow all paraeers o change a each brea dae. 4. Applcaon o U.S. Inflaon Dnacs A subsanal recen leraure s devoed o evaluang he evdence for paraeer change n e-seres odels for he pos-war U.S. nflaon rae. In parcular, Cogle and Sargen (00) argue ha he perssence of shocs o he U.S. nflaon rae have vared consderabl over he saple perod, beng que low pror o he grea nflaon and afer he Volcer dsnflaon, whle beng que hgh beween hese epsodes. The Cogle and Sargen resuls were challenged b Pvea and es (00) and Soc (00). In parcular, hese auhors argue ha evdence for shfs n perssence s no sascall sgnfcan, parcularl once one allows for shfs n he resdual varance of he odel for he nflaon rae. Oher auhors, such as Levn and Pger (00), have argued ha here are poran srucural breas n he nercep paraeer of an auoregresson for nflaon, and ha allowng for such breas s poran o properl characerze nflaon perssence. The saes n hs debae are que hgh, as he slzed facs regardng nflaon are e ercs ofen used o evaluae he plausbl of srucural acroeconoc odels. 3
4 Here we appl he Baesan odel selecon procedures descrbed above o evaluae he evdence for srucural breas n he paraeers of an auoregressve process f o he pos-war U.S. nflaon rae. We easure nflaon as he quarerl percenage change n he U.S. GDP Deflaor, easured fro he frs quarer of 953 hrough he second quarer of 005. A plo of hs daa s shown n Fgure. We are parcularl neresed n four an quesons: ) Is here srong sascal evdence of srucural change n he paraeers of an auoregresson f o he U.S. nflaon rae? ) If so, s here evdence of srucural change n condonal ean paraeers or onl n resdual varance paraeers? 3) If here s evdence of changes n condonal ean paraeers, s here evdence for changes n he perssence of he nflaon process? 4) Does allowng for srucural change n he paraeers of an auoregresson for nflaon aler our esaes of nflaon perssence? We f he followng h -order auoregressve odel wh poenal srucural breas n nercep, auoregressve paraeers, and resdual varance o he nflaon rae :........... ε ρ ρ ρ α ε ρ ρ ρ α ε ρ ρ ρ α,..,, 0 τ τ τ τ τ τ (0),...,T,
5 A e quan of neres n he odel n (0) s he perssence of, or he exen o whch an nnovaon, ε, has long-lved effecs on he level of he nflaon rae. We easure perssence va he su of he auoregressve coeffcens, denoed j j ρ ρ. As dscussed n Pvea and es (00), for < ρ, ( ) ρ / gves he area under he pulse response funcon, and s hus an nuvel appealng easure of he perssence of a e-seres process. To esae ρ drecl, we rewre (0) usng he Dce-Fuller ransforaon: ε γ γ ρ α ε γ γ ρ α ε γ γ ρ α ) ( ) ( ) (........... Δ Δ Δ Δ Δ Δ,..,, 0 τ τ τ τ τ τ (),...,T, where ( ) s j j s ρ γ and ) (0,.. ~ N d ε. In he noaon of he odel n (), we have ( ) ' ) (,...,,, Δ Δ x and ( ) ',...,,, j γ γ ρ α β,,...,. We use he followng pror specfcaons for Baesan esaon of he odel n (). For a gven value of,, and, we have he vecor of condonal ean paraeers β. The eleens of hs vecor are assgned pror ndependence, and each s gven a Gaussan pror dsrbuon wh varance equal o and ean equal o for α and ρ and 0 for γ,,...,. Each s assgned an nvered gaa pror
dsrbuon wh paraeers ν. 00 and δ. Fnall, he nu rege lengh s gven b b quarers. esuls are based on 0,000 sulaons of he Gbbs Sapler afer an nal 5000 burn-n sulaons o oban convergence. To conduc odel selecon for he odel n (), we use he preferred poseror odds procedure oulned n Secon.4. In parcular, we consder all possble cobnaons of shfs n he nercep paraeer, he auoregressve paraeers, and he resdual varance paraeer a each poenal brea dae. To econoze on he odel space consdered, we consder he auoregressve paraeers as a sngle bloc and hus assue ha hese paraeers undergo srucural breas ogeher. In addon, we exend he Baesan odel selecon procedure o allow for odel selecon over he lag lengh,. In parcular, a odel s defned b values for,, and, denoed M (, ),. We augen our pror over odels wh a fla pror over, where we consder a axu lag lengh of *. Tha s, we have: P. () N ( M (,, ) ) * * * * We hen conduc odel selecon usng he poseror odel probables, ( M,, ) ) P ( ~. As a eans of coparson, we also generae poseror odel probables fro our baselne poseror odds procedure, where we assue ha all paraeers brea a each poenal brea dae, and hus do no conduc odel selecon over, bu nsead over and onl. Fnall, for all he resuls presened below, we se * 4 and * 4. 6
Table 4 presens he poseror probables for alernave values of he nuber of srucural breas,, defned as ( M,, ) ) P ( ~. Table 4 also presens poseror probables for alernave obaned fro he baselne poseror odds procedure. The resuls for he baselne poseror odds procedure deonsrae ha here s overwhelng evdence for srucural breas, and ha 3 s he preferred nuber of srucural breas. The resuls for he baselne poseror odds procedure also sugges overwhelng evdence for srucural breas, alhough he chosen nuber of breas s four raher han hree. Nex we ove o evaluang he naure of hs srucural change. In parcular, we are neresed n evaluang he cla ha srucural breas n he nflaon rae are confned o srucural breas n he resdual varance paraeer onl, and do no exend o he condonal ean paraeers. Table 5 copares he poseror probabl of odels wh srucural breas n onl condonal varance wh he poseror probabl of odels ha conan srucural breas n condonal ean paraeers. These probables are copued boh for he preferred and baselne poseror odds procedures. As Table 5 aes clear, when odel selecon s based on he baselne procedure, he poseror probabl ha he odel conans onl breas n resdual varance donaes he poseror probabl ha he odel conans srucural breas n condonal ean paraeers. However, when odel selecon s conduced usng he preferred procedure hese probables reverse, wh odels conanng srucural breas n condonal ean paraeers donang hose wh srucural breas n onl resdual varance. Thus, hese resuls deonsrae ha odel selecon over can be crucal for evaluang he evdence for alernave pes of paraeer breas. 7
Wha s he naure of he srucural breas n condonal ean paraeers? We are parcularl neresed n wheher here s evdence n favor of shfs n auoregressve paraeers, or f he breas n condonal ean paraeers are confned o nercep shfs. An advanage of conducng odel selecon over s ha allows us o provde evdence on hs queson. In parcular, we begn b resrcng he odel space o onl hose odels ha have a srucural brea n condonal ean paraeers. We hen dvde hs odel space no hose odels ha do and do no conan srucural breas n auoregressve paraeers and consruc he poseror probabl for each class of odels. The resuls sugges ha he odels wh breas n auoregressve paraeers are gven 33% poseror probabl, whle he odels whou breas n auoregressve paraeers are gven 67% poseror probabl. Thus, whle here does no see o be srong evdence n favor of shfs n auoregressve paraeers, he daa s no speang srongl agans such shfs eher. Ths suggess ha whle here s clear evdence n favor of shfs n condonal ean paraeers, he evdence s no clear as o wheher hese shfs nclude breas n auoregressve paraeers, and hus n he perssence of he nflaon process. We now sud hose odels for whch he nuber of srucural breas s equal o he chosen value of 3 n ore deal. Table 6 conans specfcaon and esaon deals for he hghes poseror probabl odels wh 3. In parcular, each odel presened n Table 6 has a poseror probabl ha s no less han /0 h ha of he os preferred odel wh 3. Several conclusons can be drawn fro hese resuls. Frs, here are a large nuber and vare of odels n Table 6, suggesng ha he daa does no spea defnel abou he exac for of he preferred odel wh hree srucural 8
breas. Second, he ng of he srucural breas generall fall no one of wo caegores. In he frs, he srucural breas occur n he lae 960s, earl 970s and earl 980s, whle n he second he srucural breas occur n he lae 960s, earl 980s and earl 990s. Thrd, srucural breas n he nercep and resdual varance paraeers appear o be a donan feaure of he daa. In parcular, of he 5 odels presened n Table 6, all allow for a leas wo breas n resdual varance, whle 3 allow for hree breas n resdual varance. Correspondngl, 4 of he odels allow for a leas wo shfs n nercep, whle allow for hree shfs n nercep. Fourh, here s less evdence of an shfs n auoregressve paraeers. Specfcall, onl 7 of he 5 odels n Table 6 allow for a shf n he auoregressve paraeers. When he daa does no defnel selec an exac odel specfcaon, as s he case n Table 6, one would lel be hesan o base conclusons abou e paraeers on he resuls fro an one os preferred specfcaon. An advanage of he Baesan procedures eploed n hs paper s he abl o characerze e paraeers of neres whou condonng on a parcular value for,, or. For exaple, we gh be neresed n esang he su of he auoregressve coeffcens a each quarer n he saple, defned as ρ. A poseror dsrbuon for hs quan ha s averaged over poenal values for, and s gven b: f ( ρ ~ ) f ρ ~ ~. (3) ( M (,, ), ) * P( M (,, ) ) For coparson purposes, we wll also be neresed n an esae of ρ condonal on a parcular value for, gven b: 9
f ρ, ~ ) f ρ ~ ~. (4) ( ( M (,, ), )* P( M (,, ) ) Fgure presens he 5 h, 50 h and 95 h percenle of f ( ρ 0, ~ ), whch s he esae of he perssence of he nflaon process assung here have been no srucural breas. The esaed perssence of he nflaon process s que hgh, wh a edan poseror value above 0.9 and 95 h poseror percenle approachng one. Ths s conssen wh a large exsng leraure docuenng hgh nflaon perssence when easured usng a consan paraeer auoregresson over he pos-war perod (e.g. Nelson and Plosser, 98; Fuhrer and Moore, 995). Fgure 3 nsead presens he 5 h, 50 h and 95 h percenle of f ( ρ ~ ), ha s he esae of perssence allowng for srucural breas. There are a leas hree es of neres n Fgure 3. Frs of all, he edan esae of he perssence process s largel consan. Second, he edan esae of he perssence process s subsanall lower han ha obaned for he consan paraeer auoregresson presened n Fgure. For exaple, over he enre saple, he edan esae s below he 5 h percenle of he poseror for he consan paraeer auoregresson. Ths lowered perssence coes fro allowng for nercep shfs n he odel wh srucural breas. Fnall, he unceran surroundng he perssence paraeer has becoe que wde over he las wo decades, wh he 95% hghes poseror dens nerval spannng fro 0. o 0.9 oward he end of he saple. I s also neresng o characerze he poseror dsrbuon of he resdual sandard devaon a each quarer n he saple, denoed. Fgure 4 presens he 5 h, 50 h and 95 h percenle of f ( ~ ), ha s he esae of resdual sandard devaon averagng over dfferen values of,, and. Ths resdual sandard devaon has been far fro 30
consan. In parcular has vared fro a low volal rege n he 950s and uch of he 960s o a hgh volal rege fro he lae-960s o he earl 980s, before reurnng o a low volal rege begnnng n he earl 980s. In suar, hese resuls sugges ha here have been poran srucural breas n boh he condonal ean paraeers and condonal varance paraeers of an auoregresson for pos-war U.S. nflaon. We fnd ha he evdence for shfs n condonal ean paraeers are onl revealed once we conduc odel selecon over no jus he nuber of srucural breas, bu also he subse of paraeers ha brea a each brea dae, deonsrang he eprcal relevance of he odel selecon procedures developed earler n hs paper. Fnall, we fnd no srong evdence for e varaon n he perssence of U.S. nflaon. However, he esaes of perssence ha we oban are subsanall lower han hose obaned n he exsng leraure usng consan paraeer auoregressons, as hese odels gnore poran shfs n he nercep paraeer of he nflaon auoregresson. 5. Concluson We have developed a Baesan approach o odel selecon n regresson odels wh srucural breas n condonal ean and resdual varance paraeers. A novel feaure of our approach s ha does no assue nowledge of he paraeer subse ha undergoes srucural breas, bu nsead conducs odel selecon jonl over he nuber of srucural breas and he subse of he paraeer vecor ha changes a each brea dae. Sulaon experens sugges here are poenall szeable gans for brea 3
deecon o conducng odel selecon over he subse of paraeers ha undergo breas raher han spl allowng all paraeers o change. We appl he proposed odel selecon procedure o characerze possble srucural breas n condonal ean and resdual varance paraeers n an auoregressve odel for he U.S. nflaon rae. We fnd subsanal evdence for changes n boh resdual varance and condonal ean paraeers, he laer of whch s revealed onl when one conducs odel selecon over he subse of condonal ean paraeers ha undergo breas, a resul ha deonsraes he eprcal relevance of our proposed odel selecon procedure. We oban esaes of nflaon perssence ha are subsanall lower han hose obaned assung a consan paraeer auoregresson. 3
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Table Sulaon esuls: No Srucural Breas Low Perssence Moderae Perssence Hgh Perssence Preferred Procedures Poseror Odds 97.8 98.8 98.0 SIC 9.4 93.6 90.8 Baselne Procedures Poseror Odds 98.8 99.4 99.6 SIC 99.8 99.6 99.6 Noes: Ths able holds he proporon of 500 sulaons for whch he ndcaed odel selecon procedure seleced he correc value of 0 srucural breas when he daa generang process s gven b equaon (9). Preferred Procedures ndcae odel selecon s conduced over boh he nuber of srucural breas and he subse of he paraeer vecor ha changes a each brea dae. Baselne Procedures ndcae odel selecon s conduced over he nuber of srucural breas onl, wh all paraeers allowed o change a each brea dae. Low Perssence, Moderae Perssence and Hgh Perssence refer o he perssence of he regressor n equaon (9). 36
Table Sulaon esuls: One Srucural Brea Brea n β 0 Preferred Procedures Correc Nuber of Breas Sall Brea Correc Model Correc Nuber of Breas Large Brea Correc Model Poseror Odds 46. 39. 85.6 77.6 SIC 50.6 43.6 86. 80.6 Baselne Procedures Poseror Odds 6.0 NA 75.8 NA SIC 6.4 NA 63.6 NA Brea n β Preferred Procedures Poseror Odds 4.0 36. 86. 77.6 SIC 49. 43.4 88.0 8.8 Baselne Procedures Poseror Odds 4.0 NA 74.8 NA SIC 3.8 NA 6.0 NA Brea n Preferred Procedures Poseror Odds 56.0 53.0 9.4 88.4 SIC 55.4 47.8 87.6 84.0 Baselne Procedures Poseror Odds 4.0 NA 74.0 NA SIC 0. NA 68. NA Noes: Ths able holds he proporon of 500 sulaons for whch he ndcaed odel selecon procedure seleced he correc value of srucural brea (gven n he colun labeled Correc Nuber of Breas ), as well as he correc subse of he paraeer vecor ha changes a he brea dae (gven n he colun labeled Correc Model ). The daa generang process for he sulaed daa s gven b equaon (9). Preferred Procedures ndcae odel selecon s conduced over boh he nuber of srucural breas and he subse of he paraeer vecor ha changes a each brea dae. Baselne Procedures ndcae odel selecon s conduced over he nuber of srucural breas onl, wh all paraeers allowed o change a each brea dae. 37
Table 3 Sulaon esuls: Two Srucural Breas Brea n β 0 Preferred Procedures Correc Nuber of Breas Sall Breas Correc Model Correc Nuber of Breas Large Breas Correc Model Poseror Odds 55.4 4. 9.8 73.4 SIC 67. 58.0 96.0 86.0 Baselne Procedures Poseror Odds 5.0 NA 75. NA SIC. NA 49. NA Brea n β Preferred Procedures Poseror Odds 48.8 36.6 89.8 7.4 SIC 57. 47.4 93.4 8.4 Baselne Procedures Poseror Odds 5. NA 74.8 NA SIC 8.4 NA 50.0 NA Brea n Preferred Procedures Poseror Odds 65.8 60.6 94.8 8.4 SIC 69.8 59.0 97.8 88. Baselne Procedures Poseror Odds 5. NA 75.4 NA SIC 7.8 NA 75.6 NA Noes: Ths able holds he proporon of 500 sulaons for whch he ndcaed odel selecon procedure seleced he correc value of srucural breas (gven n he colun labeled Correc Nuber of Breas ), as well as he correc subse of he paraeer vecor ha changes a he brea daes (gven n he colun labeled Correc Model ). The daa generang process for he sulaed daa s gven b equaon (9). Preferred Procedures ndcae odel selecon s conduced over boh he nuber of srucural breas and he subse of he paraeer vecor ha changes a each brea dae. Baselne Procedures ndcae odel selecon s conduced over he nuber of srucural breas onl, wh all paraeers allowed o change a each brea dae. 38